Topology and Geometry Seminar

Speaker : Dr. Zhe Sun, Tsinghua University

Time:   3:30-5:30pm Wednesday, Nov 9, 2016

Place:  X2415, Xipu campus SWJTU

Title: Quantization of rank n swapping algebra for the PSL(n,R) Hitchin component (I)

Abstract:

Poisson algebra of the moduli space M of flat G-connections on a Riemann surface S is used to formulate a quantization, addressing problems in mapping class group representations, three dimensional topological invariants, conformal field theory, topological quantum field theory, Chern-Simon theory. In this talk, I will explain Atiyah-Bott-Goldman(ABG) symplectic structure on M and its relation with lattice gauge theory, Poisson-Lie group structure. The swapping algebra, introduced by F. Labourie via pairs of points on a circle, gives a Poisson algebra on the Hitchin component of M relevant to the ABG symplectic structure. I study its rank n version by taking a quotient, which I call it rank n swapping algebra. Then I Poisson embed the Fock-Goncharov coordinates(generalized Thurston shear coordinates) for cluster X_{PSL(n,R),S} moduli space into the fraction algebra of the rank n swapping algebra. Then I will give a quantization of the rank n swapping algebra, which is the first step to generalize the Kashaev's 3D invariant for the deformation parameter q at root of unit, Anderson-Kashaev's TQFT for the deformation parameter not at root of unit.